# Concluding 2010 Fall

Thanks all for participating POW actively. Here’s the list of winners:

1st prize: Kim, Chiheon (김치헌) – 수리과학과 2006학번

2nd prize: Park, Minjae (박민재) – 한국과학영재학교 (KAIST 2011학번 입학예정)

3rd prize: Jeong, Jinmyeong (정진명) – 수리과학과 2007학번.

Congratulations!

In addition to these three people, I selected one more student to receive 2 movie tickets.

Jeong, Seong-Gu (정성구) – 수리과학과 2007학번.

김치헌 (2006학번) 28 pts
박민재 (KSA) 25 pts
정진명 (2007학번) 19 pts
정성구 (2007학번) 16 pts
서기원 (2009학번) 9 pts
심규석 (2007학번) 9 pts
권용찬 (2009학번) 3 pts
정유중 (2006학번) 3 pts
진우영 (KSA) 3 pts
서영우 (2010학번) 2 pts
오상국 (2007학번) 2 pts
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# Solution: 2010-16 Number of divisors in 1 (mod 3) or 2 (mod 3)

Let n be a positive integer. Let D(n,k) be the number of divisors x of n such that x≡k (mod 3). Prove that D(n,1)≥D(n,2).

The best solution was submitted by Jeong, Seong Gu (정성구), 수리과학과 2007학번. Congratulations!

Here is his Solution of Problem 2010-16.

Alternative solutions were submitted by 정진명 (수리과학과 2007학번, +3), 김치헌 (수리과학과 2006학번, +3), 박민재 (KSA-한국과학영재학교, +3).

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# Solution: 2010-9 No zeros far away

Let M>0 be a real number. Prove that there exists N so that if n>N, then all the roots of $$f_n(z)=1+\frac{1}{z}+\frac1{{2!}z^2}+\cdots+\frac{1}{n!z^n}$$ are in the disk |z|<M on the complex plane.

The best solution was submitted by Jeong, Seong Gu (정성구), 수리과학과 2007학번. Congratulations!

Here is his Solution of Problem 2010-9.

Alternative solutions were submitted by 최홍석(화학과 2006학번, +3), 김호진(2009학번, +3), 김치헌 (수리과학과 2006학번, +3).

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# Solution: 2010-7 Cardinality

Let S be the set of non-zero real numbers x such that there is exactly one 0-1 sequence {an} satisfying $$\displaystyle \sum_{n=1}^\infty a_n x^{-n}=1$$. Prove that there is a one-to-one function from the set of all real numbers to S.

The best solution was submitted by Jeong, Seong Gu (정성구), 수리과학과 2007학번. Congratulations!

Here is his Solution of Problem 2010-7.

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# Solution: 2010-1 Covering the unit square by squares

Prove that finitely many squares on the plane with total area at least 3 can cover the unit square.

The best solution was submitted by Jeong, Seong-Gu (정성구), 수리과학과 2007학번. Congratulations!

Here is his Solution of Problem 2010-1.

Alternative solutions were submitted by 임재원 & 서기원 (2009학번, +3 -> +2, +2 each) and 권용찬 (2009학번, +2; almost correct). Thank you for participation.

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# Concluding Fall 2009

Thanks all for participating POW actively. Here’s the list of winners:

1st prize:  Jeong, Seong-Gu (정성구) – 수리과학과 2007학번

(shared) 2nd prize: Ok, Seong min (옥성민) – 수리과학과 2003학번

(shared) 2nd prize: Lee, Jaesong (이재송) – 전산학과 2005학번

Congratulations! (We have two students sharing 2nd prizes.) POW for 2010 Spring will start on Feb. 5th.

정성구 (2007학번) 35 pts
이재송 (2005학번) 10 pts
옥성민 (2003학번) 10 pts
김호진 (2009학번) 5 pts
양해훈 (2008학번) 4 pts
류연식 (2008학번) 4 pts
박승균 (2008학번) 4 pts
Prach Siriviriyakul (2009학번) 3 pts
노호성 (2008학번) 3 pts
김현 (2008학번) 3 pts
김환문 (2008학번) 3 pts
최범준 (2007학번) 3 pts
정지수 (2007학번) 3 pts
심규석 (2007학번) 3 pts
김치헌 (2006학번) 3 pts
송지용 (2006학번) 3 pts
최석웅 (2006학번) 3 pts
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# Solution: 2009-22 Integral and Limit

Evaluate the following limit:
$$\displaystyle \lim_{\varepsilon\to 0}\int_0^{2\varepsilon} \log\left(\frac{|\sin t-\varepsilon|}{\sin \varepsilon}\right) \frac{dt}{\sin t}$$.

The best solution was “again” submitted by Seong-Gu Jeong (정성구), 수리과학과 2007학번. Congratulations!

Here is his Solution of Problem 2009-22.

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# Solution: 2009-21 Rank and Eigenvalues

Let A=(aij) be an n×n matrix such that aij=cos(i-j)θ and θ=2π/n. Determine the rank and eigenvalues of A.

The best solution was submitted by Seong-Gu Jeong (정성구), 수리과학과 2007학번. Congratulations!

Here is his Solution of Problem 2009-21.

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# Solution: 2009-20 Expectation

Let en be the expect value of the product x1x2 …xn where x1 is chosen uniformly at random in (0,1) and xk is chosen uniformly at random in (xk-1,1) for k=2,3,…,n. Prove that $$\displaystyle \lim_{n\to \infty} e_n=\frac1e$$.

The best solution was submitted by Seong-Gu Jeong (정성구), 수리과학과 2007학번. Congratulations!

Here is his Solution of Problem 2009-20.

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Suppose y(x)≥0 for all real x. Find all solutions of the differential equation $$\frac{dy}{dx}=\sqrt{y}$$, y(0)=0.